I have a student who is using Euler's formula to determine the
number of panels a builder will need to construct a polyhedral dome that
is made out of 40 steel pipes and 25 connecting joints. He explains to me
that he is thinking the steel pipes are the faces and the connection
joints are vertices. He concludes that the number of panels the builder
needs equals 40+25=63. Is his reasoning correct? What follow-up question
could I ask to make certain he understands the use of Euler's formula?